On a paper by Doeblin on non-homogeneous Markov chains

1981 ◽  
Vol 13 (2) ◽  
pp. 388-401 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Field Structure ◽  
Space Decomposition

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail σ-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail σ-field structure of the chain, can be used to solve such cases.

On a paper by Doeblin on non-homogeneous Markov chains

1981 ◽  
Vol 13 (02) ◽  
pp. 388-401
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Field Structure ◽  
Space Decomposition

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail σ-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail σ-field structure of the chain, can be used to solve such cases.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (02) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

State Space Decomposition for Large Markov Chains

1995 ◽  
pp. 587-590 ◽  
Author(s):  
Maria Rieders
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Space Decomposition

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (2) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

State Space Decomposition into Cyclic Subspaces and Transformation to the Jordan Form

1984 ◽  
Author(s):  
R. P. Guidorzi ◽  
T. E. Bullock ◽  
G. Basile
Keyword(s):  
State Space ◽  
Jordan Form ◽  
Space Decomposition

Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 545-556 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Probabilistic Interpretation ◽  
Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space

2001 ◽  
Vol 33 (1) ◽  
pp. 188-205 ◽  
Author(s):  
Servet Martínez ◽  
Bernard Ycart
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Decay Rates ◽  
Hitting Times ◽  
Countably Infinite ◽  
Infinite State

THE SHANNON–MCMILLAN THEOREM FOR MARKOV CHAINS INDEXED BY A CAYLEY TREE IN RANDOM ENVIRONMENT

2017 ◽  
Vol 32 (4) ◽  
pp. 626-639 ◽  
Author(s):  
Zhiyan Shi ◽  
Pingping Zhong ◽  
Yan Fan
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Random Environment ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Countable State Space ◽  
Large Numbers ◽  
Countable State ◽  
Definition Of

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

R-theory for Markov chains on a topological state Space. II

1976 ◽  
Vol 34 (4) ◽  
pp. 269-278 ◽  
Author(s):  
David B. Pollard ◽  
Richard L. Tweedie
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Topological State
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